It may help to think about the genealogy of the sequences in question. In general, we suppose that all DNA sequences from organisms are related by a set of processes. A particularly important process relating DNA sequences is replication. This establishes a parent-child relationship between DNA sequences in one round of replication or an ancestor-descendent relationship when we view more than one round of replication.
For simplicity, consider a single bacterial cell, which replicates its genome and divides, yielding a descendent cell. This process continues for each cell and produces a population of cells over time. This population of cells (and their genomes) are related to each other by this process of replication and division. The structure of this relationship is a tree (in the mathematical sense). The vertices of the tree are the DNA sequences (or cells) and a directed edge from
vertex i to
vertex j exists if and only if the sequence represented by
vertex j is the result of a replication of the sequence represented by
In addition to the replication process which makes this tree structure relating sequences, there are mutational processes that lead to sequence differences.
There are many ways for mutations to arise before or during DNA replication. The important thing to note is that these mutations are passed on to descendants. So, a cell is either genetically identical to its parent cell, or it contains some number of mutations (insertions, deletions, or polymorphisms in the DNA) with respect to its parent.
These mutations (sequence differences) are inherited during replication, and you can think of them as "occurring" along a given edge in the tree from a parent cell to a child cell.
Now, if you consider any two DNA sequences, as well as the model just given, these two sequences are related through the replication-mutation process (the tree). If the DNA sequences are not identical, you have many ways to interpret this question of what is a mutation. It is useful to consider the most recent common ancestor of the two DNA sequences in making sense of the differences observed in the DNA. If you think of the genealogy as a tree, and you're considering two vertices in the tree, then tracing backwards towards the root of the tree from each vertex, these traces will eventually meet at another vertex. The first vertex at which they meet is the most recent common ancestor.
It is possible that mutations have occurred along both paths from the (most recent) common ancestor to the descendent sequences you're investigating. If you're investigating N sequences, there are N paths from the common ancestor on which mutations may have occurred.
All of this is to say that mutations occur with respect to this genealogical process. If you just consider 2, or N, sequences without regard to the process that generated them, then it can be meaningless or confusing to consider a particular sequence difference as a mutation with respect to another sequence under consideration. For example, if you observe a single nucleotide polymorphism between two sequences, how would you know which sequence was the result of the mutation? It is possible that one or both sequences carries a mutation event in its genealogical history since the common ancestor of both sequences.